A somewhat loose restatement of our completeness and soundness properties from above would be that we require a distribution over a space \(\Omega ^t\)  with most edges incident that are incident on the set indicated by a dictator function stay inside the set, or equivalently, dictator cuts have small edge expansion. In fact, objects with this property, namely the noisy hypercube and its variants, have been successfully used to show optimal hardness of several problems such MaxCut, UniqueGames [1]}, SmallSetExpansion [2]}. Due to the additional bias constraint, this motivates us to study the \(\mu \) -biased \((1 - \rho )\) - noisy hypercube \(\mathcal {H}_{\mu ,\rho }\) . We describe the dictatorship test as a distribution over hyperedges for \(\mathcal {H}_{\mu ,\rho }\)  in Figure REF  below.
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